Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.7% → 96.6%

Time: 25.2s

Alternatives: 7

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos \left({\left(-\sqrt[3]{M}\right)}^{3}\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (pow (- (cbrt M)) 3.0))
  (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ m n) 2.0) M) 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(pow(-cbrt(M), 3.0)) * exp(((fabs((n - m)) - l) - pow((((m + n) / 2.0) - M), 2.0)));
}

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(Math.pow(-Math.cbrt(M), 3.0)) * Math.exp(((Math.abs((n - m)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0)));
}

Julia

function code(K, m, n, M, l)
	return Float64(cos((Float64(-cbrt(M)) ^ 3.0)) * exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))))
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[Power[(-N[Power[M, 1/3], $MachinePrecision]), 3.0], $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 76.3%

      \[\leadsto \cos \color{blue}{\left(\left(\sqrt[3]{\frac{K \cdot \left(m + n\right)}{2} - M} \cdot \sqrt[3]{\frac{K \cdot \left(m + n\right)}{2} - M}\right) \cdot \sqrt[3]{\frac{K \cdot \left(m + n\right)}{2} - M}\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. pow3 76.7%

      \[\leadsto \cos \color{blue}{\left({\left(\sqrt[3]{\frac{K \cdot \left(m + n\right)}{2} - M}\right)}^{3}\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. associate-/l* 76.7%

      \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{K \cdot \frac{m + n}{2}} - M}\right)}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. fma-neg 76.7%

      \[\leadsto \cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(K, \frac{m + n}{2}, -M\right)}}\right)}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. div-inv 76.7%

      \[\leadsto \cos \left({\left(\sqrt[3]{\mathsf{fma}\left(K, \color{blue}{\left(m + n\right) \cdot \frac{1}{2}}, -M\right)}\right)}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. metadata-eval 76.7%

      \[\leadsto \cos \left({\left(\sqrt[3]{\mathsf{fma}\left(K, \left(m + n\right) \cdot \color{blue}{0.5}, -M\right)}\right)}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

4. Applied egg-rr 76.7%

   \[\leadsto \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(K, \left(m + n\right) \cdot 0.5, -M\right)}\right)}^{3}\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Taylor expanded in M around -inf 97.0%

   \[\leadsto \cos \left({\color{blue}{\left(-1 \cdot \sqrt[3]{M}\right)}}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
6. Step-by-step derivation

   1. neg-mul-1 97.0%

      \[\leadsto \cos \left({\color{blue}{\left(-\sqrt[3]{M}\right)}}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

7. Simplified 97.0%

   \[\leadsto \cos \left({\color{blue}{\left(-\sqrt[3]{M}\right)}}^{3}\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

8. Final simplification 97.0%

   \[\leadsto \cos \left({\left(-\sqrt[3]{M}\right)}^{3}\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
9. Add Preprocessing

Alternative 2: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos M \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ m n) 2.0) M) 2.0))) (cos M)))

C

double code(double K, double m, double n, double M, double l) {
	return exp(((fabs((n - m)) - l) - pow((((m + n) / 2.0) - M), 2.0))) * cos(M);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(((abs((n - m)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0))) * cos(m_1)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((Math.abs((n - m)) - l) - Math.pow((((m + n) / 2.0) - M), 2.0))) * Math.cos(M);
}

Python

def code(K, m, n, M, l):
	return math.exp(((math.fabs((n - m)) - l) - math.pow((((m + n) / 2.0) - M), 2.0))) * math.cos(M)

Julia

function code(K, m, n, M, l)
	return Float64(exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0))) * cos(M))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(((abs((n - m)) - l) - ((((m + n) / 2.0) - M) ^ 2.0))) * cos(M);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in K around 0 96.6%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. cos-neg 96.6%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 96.6%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Final simplification 96.6%

   \[\leadsto e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos M \]
7. Add Preprocessing

Alternative 3: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right| - \ell\\ \mathbf{if}\;m \leq -2.2 \cdot 10^{+56}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- n m)) l)))
   (if (<= m -2.2e+56)
     (* (cos M) (exp (+ (* (- (* m 0.5) M) (- M (* m 0.5))) t_0)))
     (* (cos M) (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m)) - l;
	double tmp;
	if (m <= -2.2e+56) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * (M - (m * 0.5))) + t_0));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m)) - l
    if (m <= (-2.2d+56)) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * (m_1 - (m * 0.5d0))) + t_0))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m)) - l;
	double tmp;
	if (m <= -2.2e+56) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * (M - (m * 0.5))) + t_0));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m)) - l
	tmp = 0
	if m <= -2.2e+56:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * (M - (m * 0.5))) + t_0))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(n - m)) - l)
	tmp = 0.0
	if (m <= -2.2e+56)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(M - Float64(m * 0.5))) + t_0)));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m)) - l;
	tmp = 0.0;
	if (m <= -2.2e+56)
		tmp = cos(M) * exp(((((m * 0.5) - M) * (M - (m * 0.5))) + t_0));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -2.2e+56], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.20000000000000016e56

   1. Initial program 64.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 86.3%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 86.3%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 96.1%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 96.1%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 96.1%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 96.1%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in n around 0 100.0%

      \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \color{blue}{\left(0.5 \cdot m - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   if -2.20000000000000016e56 < m

   1. Initial program 79.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 95.8%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 95.8%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 95.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in m around 0 80.1%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot \left(0.5 \cdot n - M\right) + {\left(0.5 \cdot n - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 80.1%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot n - M\right)}^{2} + m \cdot \left(0.5 \cdot n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 80.1%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot n - M\right) \cdot \left(0.5 \cdot n - M\right)} + m \cdot \left(0.5 \cdot n - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 83.1%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot n - M\right) \cdot \left(\left(0.5 \cdot n - M\right) + m\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 83.1%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{n \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot n - M\right) + m\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 83.1%

         \[\leadsto \cos M \cdot e^{\left(-\left(n \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{n \cdot 0.5} - M\right) + m\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 83.1%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.2 \cdot 10^{+56}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(M - m \cdot 0.5\right) + \left(\left|n - m\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -4.9 \cdot 10^{-5} \lor \neg \left(M \leq 30.5\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) + \left(\left(m - n\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -4.9e-5) (not (<= M 30.5)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (+ (* M (- M n)) (- (- m n) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -4.9e-5) || !(M <= 30.5)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(((M * (M - n)) + ((m - n) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-4.9d-5)) .or. (.not. (m_1 <= 30.5d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((m_1 * (m_1 - n)) + ((m - n) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -4.9e-5) || !(M <= 30.5)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (M - n)) + ((m - n) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -4.9e-5) or not (M <= 30.5):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(((M * (M - n)) + ((m - n) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -4.9e-5) || !(M <= 30.5))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - n)) + Float64(Float64(m - n) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -4.9e-5) || ~((M <= 30.5)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(((M * (M - n)) + ((m - n) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -4.9e-5], N[Not[LessEqual[M, 30.5]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - n), $MachinePrecision]), $MachinePrecision] + N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -4.9e-5 or 30.5 < M

   1. Initial program 78.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 98.4%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 98.4%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 84.4%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 84.4%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 84.4%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 88.6%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 88.6%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 88.6%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 88.6%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 88.6%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 88.6%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 88.6%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 88.6%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   12. Taylor expanded in M around inf 96.7%

       \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   13. Step-by-step derivation

       1. mul-1-neg 96.7%

          \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   14. Simplified 96.7%

       \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -4.9e-5 < M < 30.5

   1. Initial program 74.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 95.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 95.1%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 71.7%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 71.7%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 71.7%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 76.9%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 76.9%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 76.9%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 76.9%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 36.4%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 36.4%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 36.4%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 36.4%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-in 36.4%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(-M\right) \cdot \left(-\left(n - M\right)\right)} - \left(\ell - \left|m - n\right|\right)} \]

       2. fma-neg 36.4%

          \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{fma}\left(-M, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)}} \]

       3. add-sqr-sqrt 16.5%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{-M} \cdot \sqrt{-M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       4. sqrt-unprod 36.4%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{\left(-M\right) \cdot \left(-M\right)}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       5. sqr-neg 36.4%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\sqrt{\color{blue}{M \cdot M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       6. sqrt-unprod 19.9%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{M} \cdot \sqrt{M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       7. add-sqr-sqrt 36.4%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{M}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       8. sub-neg 36.4%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, -\color{blue}{\left(n + \left(-M\right)\right)}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       9. distribute-neg-in 36.4%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \color{blue}{\left(-n\right) + \left(-\left(-M\right)\right)}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       10. remove-double-neg 36.4%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + \color{blue}{M}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       11. add-sqr-sqrt 21.6%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

       12. fabs-sqr 21.6%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

       13. add-sqr-sqrt 58.8%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

   13. Applied egg-rr 58.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \left(m - n\right)\right)\right)}} \]
   14. Step-by-step derivation

       1. fma-undefine 58.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\left(-n\right) + M\right) + \left(-\left(\ell - \left(m - n\right)\right)\right)}} \]

       2. unsub-neg 58.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\left(-n\right) + M\right) - \left(\ell - \left(m - n\right)\right)}} \]

       3. neg-mul-1 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(\color{blue}{-1 \cdot n} + M\right) - \left(\ell - \left(m - n\right)\right)} \]

       4. +-commutative 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \color{blue}{\left(M + -1 \cdot n\right)} - \left(\ell - \left(m - n\right)\right)} \]

       5. neg-mul-1 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M + \color{blue}{\left(-n\right)}\right) - \left(\ell - \left(m - n\right)\right)} \]

       6. sub-neg 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \color{blue}{\left(M - n\right)} - \left(\ell - \left(m - n\right)\right)} \]

       7. sub-neg 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \color{blue}{\left(\ell + \left(-\left(m - n\right)\right)\right)}} \]

       8. sub-neg 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\color{blue}{\left(m + \left(-n\right)\right)}\right)\right)} \]

       9. neg-mul-1 58.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\left(m + \color{blue}{-1 \cdot n}\right)\right)\right)} \]

       10. +-commutative 58.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\color{blue}{\left(-1 \cdot n + m\right)}\right)\right)} \]

       11. distribute-neg-in 58.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \color{blue}{\left(\left(--1 \cdot n\right) + \left(-m\right)\right)}\right)} \]

       12. neg-mul-1 58.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(\left(-\color{blue}{\left(-n\right)}\right) + \left(-m\right)\right)\right)} \]

       13. remove-double-neg 58.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(\color{blue}{n} + \left(-m\right)\right)\right)} \]

       14. sub-neg 58.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \color{blue}{\left(n - m\right)}\right)} \]

   15. Simplified 58.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(M - n\right) - \left(\ell + \left(n - m\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.9 \cdot 10^{-5} \lor \neg \left(M \leq 30.5\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) + \left(\left(m - n\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -9.2 \cdot 10^{+23}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{+117}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) + \left(\left(m - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -9.2e+23)
   (* (cos M) (exp (* M n)))
   (if (<= M 3.9e+117)
     (* (cos M) (exp (+ (* M (- M n)) (- (- m n) l))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -9.2e+23) {
		tmp = cos(M) * exp((M * n));
	} else if (M <= 3.9e+117) {
		tmp = cos(M) * exp(((M * (M - n)) + ((m - n) - l)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= (-9.2d+23)) then
        tmp = cos(m_1) * exp((m_1 * n))
    else if (m_1 <= 3.9d+117) then
        tmp = cos(m_1) * exp(((m_1 * (m_1 - n)) + ((m - n) - l)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -9.2e+23) {
		tmp = Math.cos(M) * Math.exp((M * n));
	} else if (M <= 3.9e+117) {
		tmp = Math.cos(M) * Math.exp(((M * (M - n)) + ((m - n) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -9.2e+23:
		tmp = math.cos(M) * math.exp((M * n))
	elif M <= 3.9e+117:
		tmp = math.cos(M) * math.exp(((M * (M - n)) + ((m - n) - l)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -9.2e+23)
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	elseif (M <= 3.9e+117)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(M - n)) + Float64(Float64(m - n) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -9.2e+23)
		tmp = cos(M) * exp((M * n));
	elseif (M <= 3.9e+117)
		tmp = cos(M) * exp(((M * (M - n)) + ((m - n) - l)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -9.2e+23], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.9e+117], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(M - n), $MachinePrecision]), $MachinePrecision] + N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -9.2000000000000002e23

   1. Initial program 76.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 98.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 98.3%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 81.5%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 81.5%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 81.5%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 88.3%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 88.3%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 88.3%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 88.3%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 86.6%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 86.6%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 86.6%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 86.6%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   12. Taylor expanded in n around inf 40.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot n}} \]

   if -9.2000000000000002e23 < M < 3.8999999999999999e117

   1. Initial program 77.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 95.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 95.3%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 95.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 73.0%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 73.0%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 73.0%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 77.4%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 77.4%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 77.4%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 77.4%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 43.8%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 43.8%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 43.8%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 43.8%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-in 43.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(-M\right) \cdot \left(-\left(n - M\right)\right)} - \left(\ell - \left|m - n\right|\right)} \]

       2. fma-neg 43.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{fma}\left(-M, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)}} \]

       3. add-sqr-sqrt 15.8%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{-M} \cdot \sqrt{-M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       4. sqrt-unprod 35.8%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{\left(-M\right) \cdot \left(-M\right)}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       5. sqr-neg 35.8%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\sqrt{\color{blue}{M \cdot M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       6. sqrt-unprod 20.1%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{\sqrt{M} \cdot \sqrt{M}}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       7. add-sqr-sqrt 35.2%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(\color{blue}{M}, -\left(n - M\right), -\left(\ell - \left|m - n\right|\right)\right)} \]

       8. sub-neg 35.2%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, -\color{blue}{\left(n + \left(-M\right)\right)}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       9. distribute-neg-in 35.2%

          \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \color{blue}{\left(-n\right) + \left(-\left(-M\right)\right)}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       10. remove-double-neg 35.2%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + \color{blue}{M}, -\left(\ell - \left|m - n\right|\right)\right)} \]

       11. add-sqr-sqrt 20.8%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \left|\color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right|\right)\right)} \]

       12. fabs-sqr 20.8%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \color{blue}{\sqrt{m - n} \cdot \sqrt{m - n}}\right)\right)} \]

       13. add-sqr-sqrt 54.8%

           \[\leadsto \cos M \cdot e^{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \color{blue}{\left(m - n\right)}\right)\right)} \]

   13. Applied egg-rr 54.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{\mathsf{fma}\left(M, \left(-n\right) + M, -\left(\ell - \left(m - n\right)\right)\right)}} \]
   14. Step-by-step derivation

       1. fma-undefine 54.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\left(-n\right) + M\right) + \left(-\left(\ell - \left(m - n\right)\right)\right)}} \]

       2. unsub-neg 54.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(\left(-n\right) + M\right) - \left(\ell - \left(m - n\right)\right)}} \]

       3. neg-mul-1 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(\color{blue}{-1 \cdot n} + M\right) - \left(\ell - \left(m - n\right)\right)} \]

       4. +-commutative 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \color{blue}{\left(M + -1 \cdot n\right)} - \left(\ell - \left(m - n\right)\right)} \]

       5. neg-mul-1 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M + \color{blue}{\left(-n\right)}\right) - \left(\ell - \left(m - n\right)\right)} \]

       6. sub-neg 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \color{blue}{\left(M - n\right)} - \left(\ell - \left(m - n\right)\right)} \]

       7. sub-neg 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \color{blue}{\left(\ell + \left(-\left(m - n\right)\right)\right)}} \]

       8. sub-neg 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\color{blue}{\left(m + \left(-n\right)\right)}\right)\right)} \]

       9. neg-mul-1 54.8%

          \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\left(m + \color{blue}{-1 \cdot n}\right)\right)\right)} \]

       10. +-commutative 54.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(-\color{blue}{\left(-1 \cdot n + m\right)}\right)\right)} \]

       11. distribute-neg-in 54.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \color{blue}{\left(\left(--1 \cdot n\right) + \left(-m\right)\right)}\right)} \]

       12. neg-mul-1 54.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(\left(-\color{blue}{\left(-n\right)}\right) + \left(-m\right)\right)\right)} \]

       13. remove-double-neg 54.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \left(\color{blue}{n} + \left(-m\right)\right)\right)} \]

       14. sub-neg 54.8%

           \[\leadsto \cos M \cdot e^{M \cdot \left(M - n\right) - \left(\ell + \color{blue}{\left(n - m\right)}\right)} \]

   15. Simplified 54.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(M - n\right) - \left(\ell + \left(n - m\right)\right)}} \]

   if 3.8999999999999999e117 < M

   1. Initial program 73.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 91.9%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 91.9%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 94.7%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 94.7%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 94.7%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 94.7%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 94.7%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 94.7%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 94.7%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 94.7%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   12. Taylor expanded in l around inf 23.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   13. Step-by-step derivation

       1. neg-mul-1 23.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   14. Simplified 23.8%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9.2 \cdot 10^{+23}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{+117}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(M - n\right) + \left(\left(m - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+125} \lor \neg \left(\ell \leq 9.2 \cdot 10^{-39}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= l -1.65e+125) (not (<= l 9.2e-39)))
   (* (cos M) (exp (- l)))
   (* (cos M) (exp (* M n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -1.65e+125) || !(l <= 9.2e-39)) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = cos(M) * exp((M * n));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l <= (-1.65d+125)) .or. (.not. (l <= 9.2d-39))) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = cos(m_1) * exp((m_1 * n))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -1.65e+125) || !(l <= 9.2e-39)) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((M * n));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (l <= -1.65e+125) or not (l <= 9.2e-39):
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((M * n))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((l <= -1.65e+125) || !(l <= 9.2e-39))
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((l <= -1.65e+125) || ~((l <= 9.2e-39)))
		tmp = cos(M) * exp(-l);
	else
		tmp = cos(M) * exp((M * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[l, -1.65e+125], N[Not[LessEqual[l, 9.2e-39]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.65000000000000003e125 or 9.20000000000000033e-39 < l

   1. Initial program 74.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 96.7%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 96.7%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 79.8%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 79.8%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 82.3%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 82.3%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 82.3%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 82.3%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 63.5%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 63.5%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 63.5%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 63.5%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   12. Taylor expanded in l around inf 67.4%

       \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   13. Step-by-step derivation

       1. neg-mul-1 67.4%

          \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   14. Simplified 67.4%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   if -1.65000000000000003e125 < l < 9.20000000000000033e-39

   1. Initial program 79.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 96.5%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   4. Step-by-step derivation

      1. cos-neg 96.5%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   6. Taylor expanded in n around 0 75.7%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   7. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. unpow2 75.7%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      3. distribute-rgt-out 82.5%

         \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

      4. *-commutative 82.5%

         \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

      5. *-commutative 82.5%

         \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   8. Simplified 82.5%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   9. Taylor expanded in m around 0 58.8%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 58.8%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

       2. neg-mul-1 58.8%

          \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   11. Simplified 58.8%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   12. Taylor expanded in n around inf 32.6%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+125} \lor \neg \left(\ell \leq 9.2 \cdot 10^{-39}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* (cos M) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Add Preprocessing

3. Taylor expanded in K around 0 96.6%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
4. Step-by-step derivation

   1. cos-neg 96.6%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

5. Simplified 96.6%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

6. Taylor expanded in n around 0 77.7%

   \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
7. Step-by-step derivation

   1. +-commutative 77.7%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. unpow2 77.7%

      \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. distribute-rgt-out 82.4%

      \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

   4. *-commutative 82.4%

      \[\leadsto \cos M \cdot e^{\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

   5. *-commutative 82.4%

      \[\leadsto \cos M \cdot e^{\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

8. Simplified 82.4%

   \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

9. Taylor expanded in m around 0 61.1%

   \[\leadsto \cos M \cdot e^{\left(-\color{blue}{-1 \cdot \left(M \cdot \left(n - M\right)\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]
10. Step-by-step derivation

    1. associate-*r* 61.1%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-1 \cdot M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

    2. neg-mul-1 61.1%

       \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right)} \cdot \left(n - M\right)\right) - \left(\ell - \left|m - n\right|\right)} \]

11. Simplified 61.1%

    \[\leadsto \cos M \cdot e^{\left(-\color{blue}{\left(-M\right) \cdot \left(n - M\right)}\right) - \left(\ell - \left|m - n\right|\right)} \]

12. Taylor expanded in l around inf 38.4%

    \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
13. Step-by-step derivation

    1. neg-mul-1 38.4%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

14. Simplified 38.4%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))